9 EXAMPLES OF CONVOLUTIONS OF MEASURES AND FUNCTIONS 283 (i) It follows from (14.8.1.1) that the function (s9 x)^f(s'1x) is (M ® jS)-measurable. Moreover, if /? = !, the measure /•/? is bounded (13.14.4), hence \JL and /are convolvable (14.6.2) and we have But || /i * (/• p)\\ = NiOi */), which proves (14.9.2.1) in this case. For p = 2 we have, by (13.11.2.2), for any function h e g ([r ifc(x)l2d/Kx)dM(s)) also, by (13.21.9) and the left-invariance of jS, l2 djS(x) d |/i| (s) = |*d M (s) J* lAs-1*)!2 dflx) from which it follows that \a and /• yS are convolvable. Moreover, by (13.11.2.2), /(s" x*)l2 ^ H (0, so that the above relations and (13.21.9) imply J 1*)! d|/t|(s))2 ^ which proves (14.9.2.1 ) in this case. Finally, when p = -h oo, we remark that ju and the function 1 are convolvable by virtue of (14.6.3), and that ju * 1 is equal to the constant function ju(l). It follows immediately that \JL is con- volvable with every function in ££%(Q, j3) and that the inequality (14.9.2.1) is again valid (cf. (14.5.3)). (ii) The fact that the integral \f(s~lx) dfj,(s) exists for all x 6 G follows from (13.20.4). To show that ju */is continuous at a point JCQ e G, we may limit ourselves to the case where ju ^ 0. For each e > 0, there exists a compact subset K of G such that ju(()K) ^ e. Let V0 be a compact neighborhood ol x0. The function / is uniformly continuous on the compact set K~1V0, hence there exists a neighborhood V c V0 of XQ in G such that the relatior x e V impliesG, /?). The function u * f (defined almost everywhere by (14.8.2)) then